# Lesson 10

Resolvamos problemas con números decimales

## Warm-up: Observa y pregúntate: El luge (10 minutes)

### Narrative

The purpose of this Notice and Wonder is for students to consider the sport of luge and give them some numerical data that they will work with later in the lesson. The times and top speeds have been created and do not represent actual times from an event. The table is not labeled in order to encourage students to think about the meaning of the numbers.

### Launch

• Groups of 2
• Display the image.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time

### Activity

• “Discutan con su compañero lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

### Student Facing

¿Qué observas? ¿Qué te preguntas?

A B
48.532 82.13
48.561 82.75
48.626 82.81
48.634 83.07
48.708 82.80

### Activity Synthesis

• “La persona de la imagen está participando en una prueba deportiva llamada ‘luge’". Los atletas bajan en trineo por una pista de hielo empinada” // “The person in the picture is performing a sporting event called the luge. Athletes go down a steep ice track on a sled.”
• “Los números que están a la izquierda son los tiempos, en segundos, en los que varios atletas completaron el recorrido. Los números que están a la derecha son las velocidades máximas, en mph (millas por hora)” // “The numbers on the left are the times, in seconds, it took different athletes to complete the course. The numbers on the right are the maximum speed, in mph.” Consider labeling the columns of the table.
• Invite students to share what they notice about the numbers, including that in the first column, they get larger from top to bottom and they all have 3 decimal places. In the second column, there are only two decimal places and the numbers are not in increasing or decreasing order.

## Activity 1: ¿Qué tan preciso es? (20 minutes)

### Narrative

The purpose of this activity is for students to investigate a situation in which knowing a value to the thousandth place is important. Many high speed athletic events such as sprinting, cycling, downhill skiing, and the luge (studied here), are measured to the thousandth of a second in order to distinguish athletes whose finish times are very close to one another. Students examine the finishing times for the luge athletes, introduced in the warm-up, and what would happen if the times were only measured to the nearest hundredth of a second, tenth of a second, or second.

Students may use number lines to help answer the questions, but as in the previous lesson, will need to think carefully about how to label the number line.

MLR5 Co-Craft Questions. Keep books or devices closed. Display only the table, without revealing the questions. Give students 2–3 minutes to write a list of mathematical questions that could be asked about this situation, before comparing their questions with a partner. Invite each group to contribute one written question to a whole-class display. Ask the class to make comparisons among the shared questions and their own. Reveal the intended questions for this task and invite additional connections.
Engagement: Provide Access by Recruiting Interest. Optimize meaning and value. Invite students to share activities that they have competed in, participated in, or watched in which athletes’ speed determined their victory.
Supports accessibility for: Memory, Attention

### Launch

• Groups of 2
• Display a stopwatch that records time to the hundredth of a second.
• “¿Qué pueden hacer durante un segundo?” // “What can you do in one second?” (stand up, wave my hand, say my name)
• “¿Qué pueden hacer durante una décima de segundo?” // “What can you do in one tenth of a second?” (blink, type one letter)

### Activity

• 5 minutes: independent work time
• 5 minutes: partner work time
• Monitor for students who:
• use place value understanding to round the numbers
• plot the numbers on a number line to round them

### Student Facing

atleta tiempo (segundos) velocidad (millas por hora)
Atleta 1 48.532 82.13
Atleta 2 48.561 82.75
Atleta 3 48.626 82.81
Atleta 4 48.634 83.07
Atleta 5 48.708 82.80
1. ¿Cómo cambiarían los resultados de la carrera si los tiempos se registraran al segundo más cercano?
2. ¿Cómo cambiarían los resultados de la carrera si los tiempos se registraran a la décima de segundo más cercana?
3. ¿Cómo cambiarían los resultados de la carrera si los tiempos se registraran a la centésima de segundo más cercana?
4. Un atleta tuvo un tiempo de 48.85 segundos registrados a la centésima de segundo más cercana. ¿Cuáles tiempos pudo haber registrado este atleta a la milésima de segundo más cercana?
5. Un atleta tuvo un tiempo de 48.615 segundos registrados a la milésima de segundo más cercana. ¿Cuáles tiempos pudo haber registrado este atleta a la centésima de segundo más cercana?

### Activity Synthesis

• Ask previously identified students to share their responses.
• “¿Cómo cambian los tiempos de los atletas cuando los redondeamos al segundo más cercano?” // “How does rounding the times to the nearest second impact each of the athletes?” (It makes all of the times greater and impossible to distinguish. It impacts the fastest athletes the most as their times are shifted up the most.)
• “¿Cómo cambian los tiempos de los atletas cuando los redondeamos a la décima de segundo más cercana?” // “How does rounding the times to the nearest tenth of a second impact each of the athletes?” (It makes the times of the 1st, 3rd, and 4th athletes faster and the times of the 2nd and 5th athletes slower. It makes the second athlete tie for second place instead of winning second place.)
• Display the image from the solution or a student generated image.
• “¿Cómo pueden usar la recta numérica para encontrar los tiempos que se redondean a 48.85 segundos al redondear a la milésima de segundo más cercana?” // “How can you use the number line to find the times to the thousandth of a second that round to 48.85 seconds?” (I can label the tick marks and then take the ones that are closest to 48.85 and the one halfway between 48.84 and 48.85.)

## Activity 2: Comparemos velocidades (15 minutes)

### Narrative

The purpose of this activity is for students to order decimals and examine the effect of rounding on numbers continuing to use the luge context. In this activity, students investigate the top speeds of the athletes. In this case, the numbers are not listed in decreasing order because the top speeds do not correspond to the fastest times. Students order the top speeds before and after they have been rounded. Then they find a speed between two given speeds when the thousandths place is added. Since the speeds of the riders are not given to the thousandth, students will need to create values for the riders. There is one set of values students could pick, namely 82.804 and 82.805, where there is no value in between to the thousandth. If students choose these values, ask “¿Hay otras velocidades máximas posibles para estos atletas?” // “Are there different possible top speeds for these athletes?”

### Launch

• Groups of 2
• “Ahora vamos a examinar las velocidades máximas que registraron los atletas” // “We will now look at the top speeds that the different athletes recorded.”
• Highlight how fast these speeds are: most speed limits on freeways are between 65 and 75 mph and the athletes are only inches from the ice.

### Activity

• 5 minutes: independent work time
• 5 minutes: partner work time

### Student Facing

La tabla muestra las velocidades máximas, en millas por hora, de 5 atletas de luge:

Atleta 1 82.13
Atleta 2 82.75
Atleta 3 82.81
Atleta 4 83.07
Atleta 5 82.80

1. Escribe las velocidades máximas de los atletas en orden decreciente.
2. Si redondeamos las velocidades a la décima de milla por hora más cercana, ¿hay atletas que tienen la misma velocidad máxima? ¿Y si las redondeamos a la milla por hora más cercana?
3. Había un sexto atleta que fue más rápido que el corredor que registró 82.80 millas por hora, pero fue más lento que el corredor que registró 82.81 millas por hora. ¿Cuáles podrían ser las velocidades de esos 3 atletas si todas se midieran a la milésima de milla por hora más cercana?

### Activity Synthesis

• “Si medimos la velocidad a la milésima de milla por hora más cercana, ¿hay otras velocidades que pudo haber tenido el atleta que registró 82.80 millas por hora?” // “Are there different speeds the athlete at 82.80 mph could have, measured to the nearest thousandth of a mile per hour?” (yes)
• “¿Cuál es la mayor?, ¿y la menor?” // “What is the greatest? The least?” (82.804, 82.795) “Para el atleta que registró 82.81 millas por hora, ¿cuáles podrían ser su mayor velocidad y su menor velocidad si las medimos a la milésima de milla por hora más cercana?” // “What about for the athlete at 82.81? What are their fastest and slowest speeds to the thousandth of a mile per hour?” (82.814, 82.805)
• Invite students to give a possible set of top speeds, to the thousandth of a mile per hour, for athletes 3, 5, and 6.

## Lesson Synthesis

### Lesson Synthesis

“Hoy estudiamos números que representaban los tiempos y las velocidades máximas de competidores de luge y vimos cómo cambiaban al redondearlas a distintas posiciones” // “Today we studied numbers that represented times and top speeds of luge riders and how they are affected when rounded to different places.”

“¿Por qué querríamos redondear un número?” // “What are some reasons to round numbers?” (It gives a general idea of the size of a number. It’s easier to understand how big a number is when it is a round number.)

“¿Por qué podríamos preferir no redondear un número?” // “What are some reasons to keep numbers unrounded?” (If we need to know the exact size of the number then it can be important not to round it. If we want to compare two numbers, then we may need more digits to decide which is larger.)

“¿En qué se parecen redondear números decimales y redondear números enteros?” // “How is rounding decimals the same as rounding whole numbers?” (I need to think about place value and then find the closest hundredth or tenth or one just like I would look for the nearest ten, hundred, or thousand for whole numbers.)

## Student Section Summary

### Student Facing

En esta sección, representamos decimales hasta la posición de las milésimas.

La región sombreada del diagrama representa 0.542. Cada una de las 5 filas sombreadas es una décima o 0.1, cada uno de los 4 cuadrados pequeños sombreados es una centésima o 0.01, y cada uno de los 2 rectángulos pequeños sombreados es una milésima o 0.001. El número decimal 0.542 se puede representar de otras maneras:

• $$\frac{542}{1,000}$$
• quinientas cuarenta y dos milésimas
• $$(5 \times 0.1) + (4 \times 0.01) + (2 \times 0.001)$$

También podemos ubicar 0.542 en una recta numérica.

La recta numérica muestra que 0.542 está más cerca de 0.54 que de 0.55, así que 0.542 redondeado a la centésima más cercana es 0.54.